Modified Bernstein polynomials and their connectionist interpretation

The Weierstrass polynomial approximation theorem plays a central role in showing approximation capability of polynomial-based higher-order feedforward networks. The Bernstein polynomials are a family of polynomials which satisfy the Weierstrass polynomial theorem. Baldi (1991) interpreted the Bernstein polynomials in connectionist framework and argued that they can be viewed as a model for biological bell-shaped receptive fields. However, due to their strict constraint of equally-spaced input points, their connectionist interpretation has some problems in the real-world setting. In this paper, we present the modified Bernstein polynomials which have lesser constraints on the position of input points. The modified Bernstein polynomials can directly utilize given input/output data with a minor constraint on the position of the input points. We present theorems that show the approximation capability of the modified Bernstein polynomials. The relationship between the modified Bernstein polynomials and other higher-order feedforward network approaches is also discussed